0.1 Instantaneous Forward Rate
The instantaneous forward rate \(f(t,T)\) describes the forward rate in \([T,T+d]\) contracted at \(t\) when \(dt\) is small enough. All instantaneous rates are built upon the continuously compounded interest model. From the definition we hav that \[ f(t,t)=\lim_{ dt \to 0 }R(t,T, T+dt)=\lim_{ dt \to 0 }\frac{1}{dt} \log\left( \frac{p(t,T)}{p(t,T+dt)} \right)=- \frac{{\partial \log p(t,T)}}{\partial T} . \] ### Instantaneous Short Rate The instantaneous short rate \(r(t)\) describes the spot rate in \([t,t+dt]\) contracted at \(t\) when \(dt\) is small enough. Clearly plugging \(T=t\) into the instantaneous forward rate \(f(t,T)\) gives \[ r(t)=f(t,t). \] We can interpret this as the spot interest rate in a future infinitesimal time interval, so it is exactly the risk-free interest rate \(r\) we refer to in the context of options pricing.
When one opens a savings account, denote the amount of money in the account at time \(t\) as \(S_{t}^0\), then it is clear that \[ dS_{t}^0=r_{t}S_{t}^0dt, \] where \(r_{t}=r(t)\) is the short rate. As a result, \(S_{t}^0\) can be represented in terms of the short rate as \[ S_{t}^0=\exp\left( \int _{0}^tr_{s} \, ds \right), \] which tells us how to discount using the short rate.